Ecography 38: 311–320, 2015
doi: 10.1111/ecog.00722
© 2014 The Authors. Ecography © 2014 Nordic Society Oikos
Subject Editor: Timothy Keitt. Accepted 25 June 2014
Novel spatial analysis methods reveal scale-dependent spread and
infer limiting factors of invasion by Sahara mustard
Yue M. Li, Katrina M. Dlugosch and Brian J. Enquist
Y. M. Li ([email protected]), K. M. Dlugosch and B. J. Enquist, Dept of Ecology and Evolutionary Biology, Univ. of Arizona, Tucson,
AZ 85721, USA.
Multiple scale-dependent ecological processes influence species distributions. Uncovering these drivers of dynamic range
boundaries can provide fundamental ecological insights and vital knowledge for species management. We develop a transferable methodology that uses widely available data and tools to determine critical scales in range expansion and to infer
dominating scale-dependent forces that influence spread. We divide a focal geographic region into different sized square
cells, representing different spatial scales. We then used herbarium records to determine the species’ occupancy of cells at
each spatial scale. We calculated the growth in cell occupancy across scales to infer the scale dependent expansion rate. This
is the first time such a ‘box-counting’ method is used to study range expansion. We coupled this multi-scale analysis with
species distribution models to determine the range and spatial scales where suitable climate allows the species to spread, and
where other factors may be influencing the expansion. We demonstrate our methodology by assessing the spread of invasive
Sahara mustard in North America. We detect critical scales where its spread is limited (100–500 km) or unconstrained
(5–50 km) by climatic variables. Using climate-based models to assess the similarity of climate envelopes in its native and
invaded range, we find that the climate in the invaded range generally predicts the native distribution, suggesting that
either there has been little local adaptation to climate occurring since introduction or the biological interaction experienced
in the invaded range has not driven the species to occupy climatic conditions much different from its native range. Our
novel method can be broadly utilized in other studies to generate critical insights into the scale dependency of different
ecological drivers that influence the spread and distribution limits, as well as to help parameterizing predictions of future
spread, and thus inform management decisions.
Species’ distributions are shaped by a myriad of ecological
factors that fall into three general categories: physical environment, biotic interactions and dispersal (Brown et al. 1996,
Kirkpatrick and Barton 1997, McGill 2010). Disentangling
these multiple drivers of range boundaries is difficult, and can
be confounded by their inherent scale-dependency. Multiple
ecological and evolutionary processes that operate on differing spatial and temporal scales work together to set the range
boundaries of species (MacArthur 1972, Holt 2003, Gaston
2009, Holt and Barfield 2011), and consequently the distribution of species can show strong correlation with different
ecological factors when analyzed at different scales (Wiens
1989, McGill 2010, Estrada-Villegas et al. 2011).
For example, climatic variables such as annual precipitation show the strongest variation at large spatial scales,
whereas habitat variables such as soil pH vary more strongly
at finer scales. Consequently, climatic variables often appear
to be the strongest factor explaining species ranges at the
coarsest scale in space (Caughley et al. 1987, Root 1988,
Davis and Shaw 2001). Within a coarse scale, a distribution can be patchy given finer scale variation in habitat
structure, biotic interactions and dispersal limitation (GreigSmith 1979, Wethey 2002). Approaches that utilize multi-
scale analyses are essential for revealing how the influence of
specific ecological factors changes with scale (Wiens 1989).
In this study, we present a novel methodology that can
be broadly-applied to assess the scale-dependency of species’ distributions. Distribution changes – such as those that
occur in response to species introductions or climate change
– provide ecological experiments that can reveal the factors
that set range boundaries at specific scales (Holt et al. 2005,
Kinlan and Hastings 2005, Pysek and Hulme 2005, Ricklefs
2005, Sexton et al. 2009). Increasing number of studies have
assessed the distribution of invasive species across multiple
spatial scales. Most of them emphasize on how current extent
of invasion changes with spatial scales (Foxcroft et al. 2009)
and what ecological factors are associated with this scale
dependency of existing invasion extent (Catford et al. 2009,
Akasaka et al. 2012). But very few (Lonsdale 1993, Pysek
et al. 2008) have examined how historical and current rate of
range expansion change with spatial scales. It is important to
reveal the temporal dynamics of scale dependent range expansion in order to infer historical and existing ecological factors
that may influence distribution changes at specific scales.
We suspect the lack of such study is largely due to
methodological barrier. Here we illustrate the use of the
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‘box-counting’ method (Mandelbrot 1983 p. 19) to assess
the expansion rate of an invading species along a continuum
of spatial scales and across multiple decades. In particular,
we ask how the rate of decadal expansion varies across those
scales and whether there are critical scales at which the expansion rate changes dramatically. The box-counting method is
a simple and powerful tool normally used in scaling analysis
for detecting the fractal dimension of a system (Mandelbrot
1983, Morse et al. 1985, Ritchie 1998). Veldtman et al.
(2010) used this method to assess how environmental factors
affect the occupancy pattern of an invasive Acacia species.
In this study, instead of taking a static view of invasive
distribution, we extend its use to multi spatial scale analysis
of the temporal dynamics of range expansion.
To infer specific ecological factors driving scaledependent expansion, we demonstrate a focal analysis of
the importance of climate variables. We use climate based
species distribution models to predict the expansion potential
of a focal species across spatial scales. Climatic variables are
expected to restrict species distribution particularly at large
spatial scales (Whittaker et al. 2001, Pearson and Dawson
2003, McGill 2010). Here we not only evaluate this hypothesis but also attempt to quantitatively determine the range
of spatial scales where climate has the strongest influence.
Next, we infer the influence of dispersal by evaluating the
temporal change in the scale dependency of range expansion.
One can infer strong contribution by long distance dispersal
if high expansion rates prevail at large scales. Moreover, one
can infer when this contribution matters the most to
range expansion by finding periods during which large scale
expansion is most prevalent.
Our focal species is a winter annual plant Brassica tournefortii (Brassicaceae), commonly known as Sahara mustard.
It is native to Eurasia and Africa and was unintentionally
introduced to southern California in the 1920s. It has since
invaded vast areas of the southwest (Minnich and Sanders
2000), mainly in the Sonoran and Mojave Deserts. This
species grows not only on sand, sandy loam soil and rocky
hillside in the Mojave and Sonoran Deserts but also in more
mesic environments in California coasts. The distribution of
Sahara mustard in its native range covers 0–2400 m elevations (Miller and Cope 1996) and a variety of landscapes
such as desert floors, dunes, oases, desert mountains and
steppes (Maire 1965, Townsend and Guest 1980, Zohary
et al. 1980). Individual seeds germinate after major winter storms. Survived germinants have rapid phenology and
can produce seed sets as quick as in less than two months
(Marushia et al. 2012). Under favorable conditions, an individual plant can develop basal rosette leaves ∼1 m in diameter
and stem more than 80 cm tall (unpubl.), and produce ⬎ 16
000 seeds (Trader et al. 2006). Its seed coat becomes mucilaginous in contact with water, which may aid dispersal by
having wet seeds stick to objects capable of traveling long distance (Bangle et al. 2008). Sahara mustard is one of the very
few self-compatible species in the Brassica genus (Hinata and
Nishio 1980). Compared to congenic self-incompatible species, Sahara mustard produces smaller flowers with a smaller
amount of pollen grains. Its anthers face the pistils and pollinate the protruding stigma.
Its successful invasion may be a result of many of the
aforementioned attributes: tolerance of a wide range of
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environments, rapid phenology, high fecundity, aided seed
dispersal, and self-fertilization. Its decades of expansion
recorded in herbarium collections provide an opportunity
to conduct a multi-scale analysis of its spread. Moreover, its
threat to native species (Barrows et al. 2009) begs effective
management based on sound understanding of the processes
that limit its distribution. We demonstrate the use of our
novel method by asking 1) how the expansion of Sahara
mustard has varied temporally across spatial scales, 2) whether
climate limits the expansion of Sahara mustard at specific
spatial scales, and 3) how long distance dispersal affects the
scale dependency of its expansion.
Finally, we explore the potential for evolution and
biological interaction to limit invasive spread. For introduced species in particular, whether or not a species can
occupy novel environments has important implications for
predicting spread (Wiens et al. 2010). We use species distribution models constructed from the climate of non-native
range to predict the native range of invasive Sahara mustard.
If the predicted native range based on non-native distributions consists new territories not observed in its native range,
the difference may suggest that either an invasive species has
adapted to novel climates during invasion or novel biological
interactions in the invaded range has driven the species to
occupy climatic conditions that it normally cannot occupy
in the native range. If the predicted native range is only a
subset of the observed native range, the result may suggest
that either the species has not reached the climatic limits of
invasion or novel biological interactions in the invaded range
prevents the species from occupying all of its native climatic
range.
Methods
Estimating the expansion rate of Sahara mustard
across multiple spatial scales
We utilized herbarium collection records of Sahara mustard
to assess the species’ expansion over North America. Like
many other species, there is no specific long term study to
track the change in distribution of Sahara mustard. We show
that herbarium collections are useful for revealing the distribution dynamics of poorly studied species despite the potential sampling biases associated with such collections.
We searched a large number of online databases covering North America (Supplementary material Appendix 3,
Table A1). Only the following four herbarium databases had
records of Sahara mustard: the Southwest Environmental
Information Network (SEINet), Consortium of California
Herbaria (CCH), Global Biodiversity Information Facility
(GBIF) and New Mexico Biodiversity Collections Consortium.
Aggregating the herbarium records by decades, we reconstructed the distribution of the species in each decade since its
first discovery in North America. We then used this information to study the scale dependency of range expansion.
Next we assessed the robustness of the boxcounting method on a simulated diffusion growth model
(Supplementary material Appendix 1) and applied
the method to estimate the decadal expansion rate of Sahara
mustard across 7 spatial scales (5, 10, 20, 50, 100, 200, and
500 km). We selected the rectangular region that contained
all the herbarium records of Sahara mustard in North America
(25° to 38.25°N, 99.33° to 122.79°W) and evenly divided
this region into square cells of 5 ⫻ 5 km2 to 500 ⫻ 500 km2
size, corresponding to the 7 scales. We avoided scales smaller
than 5 km because imprecise or inaccurate geographic coordinates of herbarium specimens can introduce large errors to
small scale estimates. We summed the number of cells that
had herbarium records in each decade. Where the number of
occupied cells of size s in decade d is Ns, d, the expansion rate
at spatial scale of s from decade d to d ⫹ 1 was calculated
as rd ⫹ 1,s ⫽ ln(Nd ⫹ 1,s/Nd,s). Because only five records were
reported prior to 1950 (Supplementary material Appendix
3, Table A2), we applied the method to the decadal intervals
from year 1950 to 2009.
Our simulated diffusion growth model suggests that
as the spatial scale increases, the expansion rate calculated
by the box-counting method becomes more sensitive to
the placement of the grid. Fortunately, this sensitivity
can be largely eased by averaging estimates from multiple
placements (Supplementary material Appendix 1). In this
analysis, we shifted the placement of the grid in 16 directions
(π/8 radians apart) and three distances (600 m, 26 km, and
108 km) in each direction. We graphically report estimates
from all those placements and averaged them to determine
the expansion rate at each scale.
To assess how small scale imprecision in geographic coordinates affects our estimations at large scales, we perturbed
the original collection data by a normally distributed random
error. We used errors with a mean of zero and a standard
deviation of 1 km and applied the box-counting method to
this perturbed data.
Lastly, we assessed how variation in sampling efforts may
affect our analysis. The intensity of floristic surveys at different spatial scales could vary in time, confounding our analysis (e.g. fewer herbarium records within a decade could be a
result of less sampling effort). We quantified the variation in
sampling efforts by performing the same expansion analysis
on native species with presumably stable range (Delisle et al.
2003). We then subtracted this native ‘expansion’ rate (averaged over all grid placements) from the invasive expansion
rate to correct for sampling bias. We obtained the herbarium
records of three common and widely distributed native winter annual species (Plantago patagonica, Chaenactis stevioides
and Lepidium lasiocarpum) from the same databases and
within the same geographic boundary as Sahara mustard.
Using their combined records, we reconstructed the decadal
variation in herbarium sampling efforts (Supplementary
material Appendix 2) and used it as a baseline to adjust the
expansion rate of Sahara mustard.
Using MaxEnt species distribution models to
evaluate the range of Sahara mustard under suitable
climatic conditions
We predicted the range of Sahara mustard in North America
using species distribution models (SDMs) based on the
climatic conditions under which this species has been
recorded. We built our SDMs using MaxEnt (ver. 3.3.3k),
a program specifically developed for treating presence-only
records (Phillips et al. 2006, Elith et al. 2010b). Our MaxEnt
SDMs attempted to construct the species’ distribution under
the maximum influence of climatic variables we chose and
minimum influence of any other factors.
The expanding range of a non-native species does not
reflect a stable relationship with the invaded environments
(Elith et al. 2010a). This lack of equilibrium presents
challenges for modeling a potential distribution using
data from the current distribution. To address this
problem, we reduced the complexity of our model by
choosing only a few climatic variables that are most
biologically relevant and by using features and regularization parameters in MaxEnt that are more appropriate for
modeling species with an unstable range (Supplementary
material Appendix 4).
Our models used four climatic variables from the Global
Climate Database (WorldClim.org): mean temperature
of the coldest quarter (TEMPCOLDQ), precipitation of
the coldest quarter (RAINCOLDQ), annual precipitation
(RAINYEAR), and mean temperature of the warmest quarter (TEMPWARMQ), all of which are biologically relevant
to the life history of Sahara mustard (Table 1). Those variables were derived from climatic records between 1950 and
2000 (Hijmans et al. 2005), consistent with the period when
most of the occurrences of Sahara mustard were recorded
(1940–2010). The variables have a spatial resolution of
30 arcsecond (∼1 km2), which was achieved by interpolating
data from a global network of weather stations. The interpolation took into account the elevational difference in space
but ignored other local scale factors such as aspect (Hijmans
et al. 2005).
The background of our models was a polygon that consists of the majority of southwestern North America, from
which we drew 10 000 random samples. We also allowed
for further expansion by using an enlarged background
containing all lower 48 states of the U.S. and the entire
territory of Mexico. Models trained under this enlarged
background did not give qualitatively different results
(Supplementary material Appendix 4, Fig. A3 and A4).
Therefore we only report results from models based on the
standard background.
Table 1. The four climatic variables used in the species distribution
models. In line with modeling a range shifting species, we reduced
the complexity of our models by choosing a very few climatic variables that are most biologically relevant to Sahara mustard.
Climatic variable
Abbreviation
Biological relevance
Mean temperature
of the coldest
quarter
TEMPCOLDQ
Precipitation of
the coldest
quarter
RAINCOLDQ
Annual
precipitation
RAINYEAR
Mean temperature
of the warmest
quarter
TEMPWARMQ
The average temperature
condition in the growing
season of a winter annual
plant
The average resource (water)
availability in the growing
season of a winter annual
plant
The average of maximal
potential resource (water)
availability for an annual
plant
The high summer temperature required to break
seed dormancy in desert
winter annual plants
(Baskin and Baskin 2006)
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None of the climatic variables used were overly correlated with each other (Supplementary material Appendix 4,
Table A4); therefore we included all four variables in each
model.
We used two datasets of spatially distinctive records to
train our models. The first dataset included 414 occurrences
drawn from the herbarium records. The second dataset
included 2662 records drawn from both the herbaria and
invasive plant surveys (hereafter combined records). Those
surveys (Supplementary material Appendix 3, Table A1)
were conducted in the core region of its invasion (southern
California and Arizona) in the 2000s.
The first dataset reflects records acquired by relatively
equal sampling efforts across North America. The second
dataset reflects records influenced by intensive sampling
efforts in its core invaded area. Points close to the
species’ continental range limit are expected to have a
stronger influence on models based on the herbarium
records than those on the combined records. We derived
ensemble predictions from models based on both datasets
to balance the influence of occurrences at range limit and
in the range center.
We tested all of our models through 10-fold cross validation and made predictions of presence or absence based on
the logistic output of the models (Supplementary material
Appendix 4).
Comparing the climatic conditions of Sahara
mustard in its native and invaded range to infer
evolutionary constraints
We used our developed SDMs to project the range of Sahara
mustard over the continents to which it is native (Eurasia
and northern Africa). We also surveyed the existing literature
and searched the Global Biodiversity Information Facility
(GBIF) database to estimate the recorded native range
of Sahara mustard (Supplementary material Appendix 3,
Table A3). We then compared the model-projected and the
recorded native range to identify any key differences.
The literature describes detailed regional distribution and
habitat type within each country in the Middle East and
Europe, but gives very coarse scale description in northern
Africa and central Asia, often mentioning an area that covers multiple countries (Supplementary material Appendix 3,
Table A3). We didn’t find any flora describing its distribution in Morocco, Tunisia, and Burkina Faso even though the
GBIF database shows collections from those countries. We
therefore highlighted the entire territory of these countries as
potential native range of the species.
Results
Scale-dependent expansion of Sahara mustard
After corrected the bias in sampling efforts, we found that
in the 1960s the species experienced rapid expansion across
all scales (Fig. 1c). However, estimates beyond the scale of
50 km (inclusive) should be taken with caution given their
high sensitivity to grid placement (Fig. 1a). In contrast,
range expansion in the 1970s saw a substantial slowdown at
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5–200 km, but a dramatic surge at the 500 km scale reflecting range boundary expansion. This is the decade during
which the species was first recorded in western Texas, central
California and southern Utah. In the 1980s, the expansion
rate increased at the 5–20 km scale but decreased at larger
scales. Note, the increase at smaller scales is a result of correction for sampling efforts. Without the correction, the expansion rate was indicated to rather decrease at those smaller
scales (compare Fig. 1a and c). In the 1990s, the species’
range experienced a contraction across most spatial scales.
In the 2000s, the species almost stopped its expansion at
the 100–500 km scales but sustained a high rate of expansion
at the smaller scales. This is the decade in which the species
rose to prominence as a significant concern in the southwest. The herbarium collections nearly quadrupled from
76 records in the 1990s to 279 in the 2000s (Supplementary
material Appendix 3, Table A2).
With the exception of the 1960s, estimations of expansion rate of Sahara mustard were generally insensitive to
the placements of grids (Fig. 1a). As expected, sensitivity increases with the spatial scale of focus. But even at the
largest scales (200 and 500 km), there is no severe fluctuation in estimations as those seen in our simulated model
(Supplementary material Appendix 1, Fig. A1). Therefore,
the box-counting method produced reliable scale-dependent
expansion rate of Sahara mustard for the period between the
1970s and the 2000s. Estimations in the 1960s have wider
fluctuation especially at scales larger than 50 km (inclusive),
and thus are less reliable. This is probably due to lower number of sampling points.
Small scale error introduced to the geographic coordinates has minor but noticeable effect on the estimated
expansion rates across all spatial scales (Fig. 1b). However,
its effect does not qualitatively change any pattern of scale
dependency. Estimates of the expansion rate are also greatly
affected by the variation in sampling efforts (comparing
Fig. 1a and 1c).
Predicted range of Sahara mustard in North America
under its suitable climate
The range boundaries of Sahara mustard predicted by our
climate-based SDMs generally agreed with the recorded
boundaries at the regional scale (Fig. 2), indicating little room
for further regional scale expansion of this species except in
the large area of the Central Valley, California. According to
our models, Sahara mustard is likely to occur in areas with
mean temperature in the coldest quarter between 3.9 and
17°C, mean temperature of the warmest quarter between
15.7 and 36°C, precipitation of the coldest quarter between
16 and 302 mm, and annual precipitation between 46 and
657 mm (Supplementary material Appendix 4, Fig. A5).
All SDMs performed well according to their crossvalidation test scores (Supplementary material Appendix
4, Table A6). Models trained by herbarium records (more
influenced by points at the range limit) predicted a
more extended range (green areas in Fig. 2) that includes
more areas in northern Mexico, Baja California, central and
coastal California and a small region in northwestern Florida
(not shown in figure). In comparison, models trained by
Figure 1. Decadal expansion rates of Sahara mustard from the 1960s to the 2000s at spatial scales of 5–500 km, as estimated from
(a) original herbarium data and (b) original data perturbed by a normally distributed random error with a standard deviation of 1 km. ‘⫹’
indicates each estimate from a single grid placement. Circles indicate a mean value averaged over estimates from all grid placements.
(c) Expansion rate adjusted for variation in sampling efforts by subtracting the mean value by an estimate for native species.
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combined records (more influenced by points in the core
invaded area) predicted a more thorough coverage by the
species over the Mojave Desert (yellow areas in Fig. 2).
Agreement of climatic conditions between native
and invaded range
The recorded native range of Sahara mustard generally agrees
with the projection based on the climatic conditions in its
North American range (Fig. 3), suggesting that either this
species has not adapted to novel climatic conditions during
invasion or biological interactions in the invaded range has
not driven the species to occupy new climatic conditions. The
SDMs trained by herbarium records predicted a native range
that generally extends more southward into the drier regions
in Sahara, on the Arabian Peninsula and in central Asia (green
areas in Fig. 3), whereas models trained by combined records
in North America predicted more coverage in Turkmenistan
(yellow areas in Fig. 3), which has drier and colder winters
than regions further south. The models predict that Sahara
mustard is likely to occur on its native continents in areas with
mean temperature of the coldest quarter between –2.3 and
20.6°C, mean temperature of the warmest quarter between
14.8 and 36.7°C, precipitation of the coldest quarter between
0 and 449 mm, and annual precipitation between 0 and 782
mm (Supplementary material Appendix 4, Fig. A5).
There are some mismatches between the model projected
and the recorded native range. No records suggest the species’ presence in the Nejd region of Saudi Arabia and in
Turkmenistan. But our models, even those more influenced
by points in the core invaded range predicted its presence in
Figure 2. Distribution of Sahara mustard in North America
predicted by climate-based SDMs. One model used only herbarium records (red dots) and the other, herbarium and invasive plant
survey records (blue triangles) combined. Each ensemble shows the
area predicted by both models (peach) and by each model alone
(green or yellow). An additional small area predicted by the model
in Dixon County, Florida is not shown.
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Figure 3. Projected distribution of Sahara mustard in its native
range using SDMs based on the climate in its invaded range in
North America. Striped areas represent its native range recorded in
the literature and the GBIF database.
those regions. None of the models predicted its presence in
Burkina Faso, but the species has been found in this country
according to GBIF.
Discussion
Climatic constraints on the spread of Sahara mustard
and their spatial extent
Our results indicate that the spread of Sahara mustard in
North America is indeed spatially scale-dependent and that
climate is limiting its spread at regional scales. The expansion
of the species at scales beyond 100 km declined to a negligible
level in the 2000s (Fig. 1), suggesting a cessation of further
expansion of its range boundaries at those scales. The current
boundaries of its regional distribution largely agree with those
predicted by climate based SDMs (Fig. 2), supporting the
role of climate in restricting the spread of the species at those
large scales (100–500 km). This scale-dependent pattern in
our study supports the common view that long-term climate
shapes species distributions at relatively large scales. Although
this is not a surprising finding, our method is unique in its
ability of defining a quantitative extent of the scales of influence. Of course, we cannot rule out the possibility that the
expansion is also limited by other unexamined factors that
vary strongly at the 100–500 km scales.
Our results also reveal that any constraints imposed by
long-term climate vanish at smaller scales. Rapid expansion
continued at 5–50 km scales in the 2000s with a rate generally higher than in the 1970s–1990s (Fig. 1). Our climate
based SDMs indicate ample space for the species to further
occupy inside its regional climatic envelope, suggesting that
long-term climate is not limiting the species’ expansion at
local to intermediate scales.
The influence of dispersal on the spread of Sahara
mustard
Dispersal is generally expected to drive rates of expansion
in invasive species (Kot et al. 1996, Phillips et al. 2010),
and Sahara mustard in particular appears to have reached its
climatic limits rapidly as a result of long-distance movement.
At the 500 km scale, aside from the less reliable estimate
in the 1960s, the expansion rate was consistently high in
both the 1970s and the 1980s, supporting the view that
long distance dispersal at the scale of hundreds of kilometers allowed the species to quickly reach its regional climatic
envelope in merely a few decades. Moreover, expansion rates
at larger scales were often greater than or equivalent to those
at smaller scales between the 1960s and 1980s, suggesting
long distance movement that did not rely on sequential population establishment along a shorter dispersal path. Those
dispersal leaps could allow Sahara mustard to rapidly extend
its range over spatially heterogeneous environments, skipping hostile territories and paving the way for future space
filling within its range limit. Its ongoing rapid expansion at
the 5–50 km scales also suggests low dispersal limitation at
those smaller scales.
Processes that drive the spatiotemporal dynamics
of Sahara mustard expansion
The expansion of an invasive species has three phases: a
quiescent period between initial colonization and rapid range
expansion (the lag phase), a period of rapid range expansion
(the spreading phase), and a stage of little further expansion
(the ceased phase) (Pysek and Hulme 2005). Scale dependency of invasive spread means that any of the three phases
may occur simultaneously when observed at different spatial
scales (Pysek and Hulme 2005, Arim et al. 2006). In our
case, the expansion of Sahara mustard in the 1960s–1980s
was in the spreading phase across all scales. In the 2000s, the
species was in the spreading phase at the 5–50 km scales but
in the ceased phase at larger scales. Climatic constraints can
explain the ceased phase, while the availability of fine scale
niche space allowed the species to further expand at smaller
scales.
Our results also showed that Sahara mustard had experienced two distinctive lag phases. The first lag phase occurred
from the 1920s to the 1940s, during which there was a 14 yr
gap between its first and second recorded occurrence (1927
and 1941). The second lag phase occurred in the 1990s, during which expansion was limited across all scales.
An invasive species can experience multiple lag phases due
to different causes (Wangen and Webster 2006). In our case,
the first lag phase may reflect the time required by Sahara
mustard to overcome the problems commonly faced by small
founding populations such as loss of genetic diversity, Allee
effects, and demographic stochasticity (Crooks and Soulé
1999, Sakai et al. 2001). But it might also reflect the time
between the first introduction and a later introduction of a
more invasive form of the species. There is some anecdotal
evidence that the first introduced form of Sahara mustard,
which can still be found near Mecca, California, is smaller
and more gracile than those spreading in North America
and is more specialized in disturbed soils (M. Dimmit pers.
comm.).
The second lag phase, during which the species was well
established in North America, was most likely caused by
factors other than those that hindered its early spread. A
likely candidate is a period of less-favorable environmental
conditions, such as low water availability (a key resource for
desert annuals). We examined whether reduced cold (growing) season precipitation was associated with this lag period.
Our analysis shows that the cold season precipitation averaged
over the 1990s was in fact above the long-term mean in areas
where Sahara mustard achieved low (but positive) local-scale
expansion (Supplementary material Appendix 5), contradicting a simple explanation of low resource availability.
Plasticity vs adaptation as a process for successful
invasion
An introduced species can become a widespread invader by
either having low genetic diversity and a ‘general purpose’
genotype or rapidly adapting to its various new environments (Parker et al. 2003). A species with a general purpose
genotype tolerates a wide range of environments and can
grow in a multitude of climates and habitats (Baker 1965).
Sahara mustard, which rapidly spread to cover a broad range
of climatic and habitat conditions is a strong candidate for
carrying such a plastic genotype. Sahara mustard reproduces
primarily by self-fertilization, which is a strategy often associated with maintaining a general purpose genotype (Baker
1965), and limiting the capacity of new populations to
rapidly adapt to local conditions (Barrett et al. 2008). Those
ideas are supported by the general agreement between the
climatic conditions over its native and invaded range, suggesting that the species’ strategies for coping with climatic
variation are through well-preserved plasticity rather than
novel adaptation. Therefore, its invasion may reflect niche
conservatism (Wiens et al. 2010).
Nevertheless, just how much self-fertilization limits
Sahara mustard’s genetic variability is an open question. Its
native populations in Iraq were found to retain the potential for outcrossing (Al-Shehbaz 1977). Used as female parents, Sahara mustard was also able to artificially hybridize
with other Brassicaceae species (Hinata et al. 1975). These
opportunities for genetic recombination leave open the possibility that traits may evolve in invading Sahara mustard,
with potential impacts on its distribution. Such genetic
changes are particularly likely if repeated introductions are
made from different parts of the native range, introducing new adaptive variation into the invasion (Ellstrand and
Schierenbeck 2000, Dlugosch and Hays 2008, Dlugosch
and Parker 2008), though in this case Sahara mustard
appears to already have invaded most habitats that reflect
the breadth of climates occupied in its native range.
Management implication for Sahara mustard and
other invasive species
The climatic limits on the regional expansion of Sahara mustard and the species’ continuing spread at 5–50 km scales
suggest that control efforts should focus on containing ongoing local expansion. Our SDM prediction highlights areas
where climate provides ample room for local-scale expansion
of the species. Prevention, early detection and eradication
should focus on those high risk areas to slow down its local
expansion. Nevertheless, regional scale monitoring is needed
317
to detect any new shifts of its range boundaries under the
changing global climate.
The rapid spread of this species is fueled by effective
long distance dispersal, which may be carried out by vehicle
transportation. Early detection efforts should focus on suitable
habitats with strong linkage, such as those accessible by roads.
Strength and novelty of our approach
Using widely available data from herbarium collections, the
box-counting method successfully revealed the scale dependent spread of an invasive species, whose invasion history was
not closely monitored. The method can be readily adopted
to study the scale dependent expansion of many other species, for which the only data for interpreting range dynamics
are from long term herbarium collections. Some studies have
used the box-counting method to assess the spatial structure
of species distribution across multiple spatial scales (Foxcroft
et al. 2009, Veldtman et al. 2010, Akasaka et al. 2012),
but all have focused only on the current distribution and
neglected the temporal range dynamics. Our work presents
the first case of applying the box-counting method to
studying scale-dependent range expansion.
Our analysis effectively uncovered the scale dependency
of an invasive spread. Quantifying such scale dependency
and its contributing factors not only fosters understanding of species distribution but is also essential for effective
control of any biological invasion. By highlighting critical scales where an invasion is contained or unconstrained
due to specific factors, one can direct the management
efforts to the appropriate range and scales where invasion
is to continue. However, so far few studies have examined expansion rate of invasive species using multi-scale
approaches (Lonsdale 1993, Pysek et al. 2008). Those that
did use the approach limited their investigation to a very
few discrete scales (e.g. local to regional to continental
scale in Pysek et al. 2008) with a large range of intermediate scales missing in-between. Failing to assess a continuous range of scales can result in missing critical scales
where major change in the expansion rate takes place.
Due to these limitations, we have poor knowledge of the
prevalence and general pattern of the scale dependency of
invasive expansion. Less do we know the factors that lead
to such scale dependency (Pysek and Hulme 2005). The
methods demonstrated in our study provide a powerful
tool for filling this research gap in order to better inform
the management of many invasive species.
Caveats of our approach
Herbarium collections provide inexpensive and widely
available long-term data for inferring species distributions.
However, using those data can be challenging because they
were not collected with the intention to discover multi-scale
distribution patterns. We have incorporated in our methods
means to reduce the influence by potential sampling biases.
Nevertheless, several sources of error may still influence this
type of analysis.
First, many of the herbarium collection records have
imprecise or inaccurate geographic coordinates. We showed
318
that such errors at smaller scale could quantitatively impact
estimates of expansion at larger scales, though they would be
unlikely to alter patterns qualitatively.
Second, various spatial sampling biases could affect our
analysis. Our method used the ‘expansion’ rate of native species with assumed stable range to correct for difference in
spatial sampling efforts between decades. However, if the
nature of spatial sampling bias is different between invasive
and native species, our method will not be able to adequately
correct the bias specific to the invader. For instance, if
herbarium collectors paid more attention to a rapidly
expanding invasive species more recently but meanwhile
consistent attention to the natives, our method will overestimate the expansion rate for the invasive.
It is also possible that collectors might avoid collecting
species in the same local area over time. This sampling bias
does not necessarily affect the accuracy of our estimates
because our method relies on measuring the change in total
number of occupied cells in each time step rather than tracking the geographic movement of those cells. Making a first
collection in one cell and the second in a different cell will
not change the total number of occupied cells between the
two time steps and thus will not change the estimated expansion rate at the scale represented by those cells. However,
such method will affect our estimates if it forces multiple
collections to concentrate in one cell at one time and to
diverge to different cells at another time. Also, if collectors
specifically avoid returning to the same places, those locations will not be recorded in the following decade. Therefore,
there may be a smaller number of occupied cells than there
should be, leading to an underestimate of the expansion rate.
Our method would not be able to fully correct the bias,
if such bias is different between the invasive and the native
collections.
Our method assumes equal sampling efforts over space.
It is evident that few records of Sahara mustard exist in Mexico
and southern Central Valley of California despite the prediction of high likelihood of its presence by our SDMs (Fig. 2).
Scarcity of records in Mexico may be a direct result of lower
collection effort or the lack of digitally available data from
Mexican herbaria. Scarcity of records in southern Central
Valley of California might be due to limited habitat. Most
of the valley has been developed for industrial scale farming. Sahara mustard might lack habitat there and herbarium
collectors might avoid this region due to the lack of natural
plant habitat. A search for all specimens of Asteraceae and
Brassicaceae recorded in California Consortium of Herbaria
show that the number of records from the three counties
in the valley (Kern, Merced and San Joaquin County) was
orders of magnitude lower than those in adjacent counties
with more abundant natural plant habitat. Any large disparity in sampling efforts between regions will result in biased
estimates of expansion rate across the regions. In our case, we
know very little how the historical and current distribution
of Sahara mustard in Mexico and Central Valley, California
would have affected the estimated expansion rate across
North America. Fortunately, the majority of its predicted
range was relatively well sampled by herbarium collectors
(Fig. 2).
Spatial sampling bias will also affect the prediction by
MaxEnt SDMs, which assume equal sampling efforts over
the entire area of interest (Phillips et al. 2009). If herbarium
records are spatially biased towards better surveyed area for
the species (e.g. areas where an invasive species is a serious
concern), less surveyed areas will have less influence on the
SDMs, resulting in higher omission errors in those areas.
Indeed, our SDMs using southwestern North America as
the background fail to predict the presence of Sahara mustard in some of its recorded eastern and southern range
boundaries. Fortunately, our models using North America
as the background do include those areas as potential range
for Sahara mustard (Supplementary material Appendix 4,
Fig. A3). The latter models use a larger background with
a wider range of values in explanatory variables. Values
from the recorded range boundaries become less extreme
in models using this enlarged background. An alternative
solution to minimize such problem caused by spatial sampling bias is using spatial filtering to reduce the number of
occurrence records in oversampled areas (Kramer-Schadt
et al. 2013).
Finally, our analysis only indirectly infers the limiting
factors that drive the scale dependency of invasive range
expansion. Our methods look for correlation between
certain ecological factors and observed scale dependency
of range expansion. It provides useful insight for designing
future studies that can directly test mechanisms that drive
or limit range expansion through manipulative experiments
conducted at various scales.
Conclusions
In this study we have demonstrated the validity of a novel
methodology that can be broadly applied to uncover the
scale dependency of invasive range expansion and its contributing ecological and evolutionary factors. Large online
herbarium databases provide free and accessible data to
which the box-counting method can be applied to quantify scale-dependent spread of many invasive plant species
(after adjusting for variation in sampling efforts). Coupling
this multi-scale analysis with MaxEnt SDMs, one can
effectively determine on what scales the variables chosen
in the models are limiting or contributing to the spread.
Controlling efforts can then focus on the scales where the
invasion is not constrained and areas where ecological factors are still favoring spread. Moreover, using SDMs based
on variables in the invasive range to project its native distribution allows inference of the adaptation of an invasive
species to novel environmental conditions. Such knowledge is essential for predicting the risk of further invasion
powered by evolution.
Acknowledgements – John Donoghue II, Irena Simova and Jin Wu
guided us to cross the mysterious terrain of ArcGIS. Naia MoruetaHolme circled many MaxEnt pitfalls that might have trapped us.
Cal-IPC and Cameron Barrows shared important data of Sahara
mustard recorded in the Californian deserts. Mara MacKinnon and
Kimberly Byers spent months finding the latitude and longitude
coordinates of hundreds of herbarium records. We thank Cameron
Barrows, Aaryn Olsson, and Peter Chesson, who inspired critical
improvement on the manuscript. We also thank all the herbarium
collectors for documenting the history of Sahara mustard invasion.
Funding was proved to YML by the US Marine Corps Air Stations,
Yuma and Luke Air Force.
References
Akasaka, M. et al. 2012. Invasive non-native species attributes
and invasion extent: examining the importance of grain size.
– J. Veg. Sci. 23: 33–40.
Al-Shehbaz, I. A. 1977. Protogyny in the Cruciferae. – Syst. Bot.
2: 327–333.
Arim, M. et al. 2006. Spread dynamics of invasive species. – Proc.
Natl Acad. Sci. USA 103: 374–378.
Baker, H. G. 1965. Characteristics and modes of origin of weeds.
– In: Baker, H. G. and Stebbins, G. L. (eds), The genetics of
colonizing species. Academic Press, pp. 147–169.
Bangle, D. N. et al. 2008. Seed germination of the invasive
plant Brassica tournefortii (Sahara mustard) in the Mojave
Desert. – West. N. Am. Nat. 68: 334–342.
Barrett, S. C. H. et al. 2008. Plant reproductive systems and evolution during biological invasion. – Mol. Ecol. 17: 373–383.
Barrows, C. W. et al. 2009. Effects of an invasive plant on a desert
sand dune landscape. – Biol. Invasions 11: 673–686.
Baskin, C. C. and Baskin, J. M. 2006. The natural history of soil
seed banks of arable land. – Weed Sci. 54: 549–557.
Brown, J. H. et al. 1996. The geographic range: size, shape, boundaries, and internal structure. – Annu. Rev. Ecol. Syst. 27:
597–623.
Catford, J. A. et al. 2009. Reducing redundancy in invasion ecology by integrating hypotheses into a single theoretical framework. – Divers. Distrib. 15: 22–40.
Caughley, G. et al. 1987. Kangaroos and climate: an analysis of
distribution. – J. Anim. Ecol. 56: 751–761.
Crooks, J. A. and Soulé, M. E. 1999. Lag times in population
explosions of invasive species: causes and implications. – In:
Sandlund, O. T. et al. (eds), Invasive species and biodiversity
management. Kluwer, pp. 103–125.
Davis, M. B. and Shaw, R. G. 2001. Range shifts and
adaptive responses to Quaternary climate change. – Science
292: 673–679.
Delisle, F. et al. 2003. Reconstructing the spread of invasive
plants: taking into account biases associated with herbarium
specimens. – J. Biogeogr. 30: 1033–1042.
Dlugosch, K. M. and Hays, C. G. 2008. Genotypes on the move:
some things old and some things new shape the genetics
of colonization during species invasions. – Mol. Ecol. 17:
4583–4585.
Dlugosch, K. M. and Parker, I. M. 2008. Invading populations of
an ornamental shrub show rapid life history evolution despite
genetic bottlenecks. – Ecol. Lett. 11: 701–709.
Elith, J. et al. 2010a. The art of modelling range-shifting species.
– Methods Ecol. Evol. 1: 330–342.
Elith, J. et al. 2010b. A statistical explanation of MaxEnt for
ecologists. – Divers. Distrib. 17: 43–57.
Ellstrand, N. C. and Schierenbeck, K. A. 2000. Hybridization as
a stimulus for the evolution of invasiveness in plants? – Proc.
Natl Acad. Sci. USA 97: 7043–7050.
Estrada-Villegas, S. et al. 2011. Climate, habitat, and species
interactions at different scales determine the structure of a
Neotropical bat community. – Ecology 93: 1183–1193.
Foxcroft, L. C. et al. 2009. Patterns of alien plant distribution at
multiple spatial scales in a large national park: implications for
ecology, management and monitoring. – Divers. Distrib. 15:
367–378.
Gaston, K. J. 2009. Geographic range limits: achieving synthesis.
– Proc. R. Soc. B 276: 1395–1406.
Greig-Smith, P. 1979. Pattern in vegetation. – J. Ecol. 67:
755–779.
Hijmans, R. J. et al. 2005. Very high resolution interpolated
climate surfaces for global land areas. – Int. J. Climatol. 25:
1965–1978.
Hinata, K. and Nishio, T. 1980. Self-incompatibility in Crucifers.
– In: Tunoda, S. et al. (eds), Brassica crops and wild allies:
319
biology and breeding. Japan Scientific Societies Press,
pp. 223–234.
Hinata, K. et al. 1975. Interspecific crossability in the tribe
Brassicaceae with special reference to the self-incompatibility.
– Tohoku J. Agric. Res. 25: 58.
Holt, R. D. 2003. On the evolutionary ecology of species’ ranges.
– Evol. Ecol. Res. 5: 159–178.
Holt, R. D. and Barfield, M. 2011. Theoretical perspectives on
the statics and dynamics of species’ borders in patchy environments. – Am. Nat. 178: S6–S25.
Holt, R. D. et al. 2005. Theories of niche conservatism and evolution. – In: Sax, D. F. et al. (eds), Species invasions: insights into
ecology, evolution, and biogeography. Sinauer Associates, pp.
259–290.
Kinlan, B. P. and Hastings, A. 2005. Rates of population spread
and geographic range expansion: what exotic species tell us.
– In: Sax, D. F. et al. (eds), Species invasions: insights into
ecology, evolution, and biogeography. Sinauer Associates,
pp. 381–419.
Kirkpatrick, M. and Barton, N. H. 1997. Evolution of a species’
range. – Am. Nat. 150: 1–23.
Kot, M. et al. 1996. Dispersal data and the spread of invading
organisms. – Ecology 77: 2027–2042.
Kramer-Schadt, S. et al. 2013. The importance of correcting for
sampling bias in MaxEnt species distribution models. – Divers.
Distrib. 19: 1366–1379.
Lonsdale, W. M. 1993. Rates of spread of an invading species
– Mimosa pigra in northern Australia. – J. Ecol. 81: 513–521.
MacArthur, R. 1972. Geographical ecology: patterns in the
distribution of species. – Harper and Row.
Marie, R. 1965. Flore de L’Afrique du Nord. – Paul Lechevalier,
Paris.
Mandelbrot, B. B. 1983. The fractal geometry of nature. – W. H.
Freeman.
Marushia, R. G. et al. 2012. Phenology, growth, and fecundity as
determinants of distribution in closely related nonnative taxa.
– Invasive Plant Sci. Manage. 5: 217–229.
McGill, B. J. 2010. Matters of scale. – Science 328: 575–576.
Miller, A. G. and Cope, T. A. 1996. Flora of the Arabian Peninsula
and Socotra. – Edinburgh Univ. Press.
Minnich, R. A. and Sanders, A. C. 2000. Brassica tournefortii
Gouan. – In: Bossard, C. C. et al. (eds), Invasive plants
of California’s wildlands. Univ. of California Press,
pp. 68–72.
Morse, D. R. et al. 1985. Fractal dimension of vegetation and
the distribution of arthropod body lengths. – Nature 314:
731–733.
Parker, I. M. et al. 2003. An evolutionary approach to understanding the biology of invasions: local adaptation and generalpurpose genotypes in the weed Verbascum thapsus. – Conserv.
Biol. 17: 59–72.
Pearson, R. G. and Dawson, T. P. 2003. Predicting the impacts of
climate change on the distribution of species: are bioclimate
Supplementary material (Appendix ECOG-00722 at ⬍www.
ecography.org/readers/appendix⬎). Appendix 1–5.
320
envelope models useful? – Global Ecol. Biogeogr. 12:
361–371.
Phillips, S. J. et al. 2006. Maximum entropy modeling of species
geographic distributions. – Ecol. Model. 190: 231–259.
Phillips, S. J. et al. 2009. Sample selection bias and presence-only
distribution models: implications for background and pseudoabsence data. – Ecol. Appl. 19: 181–197.
Phillips, B. L. et al. 2010. Evolutionarily accelerated invasions: the
rate of dispersal evolves upwards during the range advance of
cane toads. – J. Evol. Biol. 23: 2595–2601.
Pysek, P. and Hulme, P. E. 2005. Spatio-temporal dynamics of
plant invasions: linking pattern to process. – Ecoscience 12:
302–315.
Pysek, P. et al. 2008. Comparing the rate of invasion by Heracleum
mantegazzianum at continental, regional, and local scales.
– Divers. Distrib. 14: 355–363.
Ricklefs, R. E. 2005. Taxon cycles: insights from invasive species.
– In: Sax, D. F. et al. (eds), Species invasions: insights
into ecology, evolution, and biogeography. Sinauer Associates,
pp. 165–199.
Ritchie, M. 1998. Scale-dependent foraging and patch choice in
fractal environments. – Evol. Ecol. 12: 309–330.
Root, T. 1988. Environmental factors associated with avian distributional boundaries. – J. Biogeogr. 15: 489–505.
Sakai, A. K. et al. 2001. The population biology of invasive species.
– Annu. Rev. Ecol. Syst. 32: 305–332.
Sexton, J. P. et al. 2009. Evolution and ecology of species range
limits. – Annu. Rev. Ecol. Evol. Syst. 40: 415–436.
Trader, M. R. et al. 2006. Seed production by the non-native
Brassica tournefortii (Sahara mustard) along desert roadsides.
– Madrono 53: 313–320.
Townsend, C. C. and Guest, E. 1980. Flora of Iraq. – Ministry of
Agriculture and Agrarian Reform, Baghdad.
Veldtman, R. et al. 2010. Using scale–area curves to quantify the
distribution, abundance and range expansion potential of an
invasive species. – Divers. Distrib. 16: 159–169.
Wangen, S. R. and Webster, C. R. 2006. Potential for multiple lag
phases during biotic invasions: reconstructing an invasion of
the exotic tree Acer platanoides. – J. Appl. Ecol. 43: 258–268.
Wethey, D. S. 2002. Biogeography, competition, and microclimate: the barnacle Chthamalus fragilis in New England.
– Integr. Comp. Biol. 42: 872–880.
Whittaker, R. J. et al. 2001. Scale and species richness: towards a
general, hierarchical theory of species diversity. – J. Biogeogr.
28: 453–470.
Wiens, J. A. 1989. Spatial scaling in ecology. – Funct. Ecol. 3:
385–397.
Wiens, J. J. et al. 2010. Niche conservatism as an emerging principle in ecology and conservation biology. – Ecol. Lett. 13:
1310–1324.
Zohary, M. et al. 1980. Conspectus Florae Orientalis: an annotated
catalogue of the flora of the Middle East. – The Israel Academy
of Sciences and Humanities, Jerusalem.